The author wants her students to see any new mathematics--fractions, negative numbers, algebra--as logical extensions of what they already know. This article describes two students' efforts to make sense of their conflicting interpretations of 1/2 × -6, both of which were compelling and logical to them. It describes how discussion, constructing…

In the set of fractal activities described in this article, students will accomplish much more than just creating a fun set of cards that simply resemble an art project. Goals of this activity, designed for an algebra 1 class, are to encourage students to generate data, look for and analyze patterns, and create their own models--all from a set of…

One of the primary expectations that the author has for her students is for them to develop greater independence when solving complex and unique mathematical problems. The story of how the author supports her students as they gain confidence and independence with complex and unique problem-solving tasks, while honoring their expectations with…

A vectorial representation of the full sequence of events occurring during the 2D-NMR heteronuclear single-quantum correlation (HSQC) experiment is presented. The proposed vectorial representation conveys an understanding of the magnetization evolution during the HSQC pulse sequence for those who have little or no quantum mechanical background.…

In this article we examine 2 x 2 first-order systems of ordinary differential equations and show how to identify separatrices for phase plane portraits when the system has a saddle point critical value. We describe how to use a computer algebra system to generate trajectories from contour plots, when possible, and determine the equation of the…

Students enjoy origami and like making everything from paper cranes to footballs out of small, colorful squares of paper. They can invent their own shapes and are intrigued by the polyhedrons that they can construct. Paper folding is fun, but where is the math? Unless teachers develop lessons that address mathematical objectives, origami could be…

This module was initially developed for a course in applications of mathematics in biology. The objective of this lesson is to investigate how the allele and genotypic frequencies associated with a particular gene might evolve over successive generations. The lesson will discuss how the Hardy-Weinberg model provides a basis for comparison when…

This book provides a landscape of learning that helps teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. It identifies the models, contexts, and landmarks that facilitate algebraic thinking in young students and provides insightful and practical methods for teachers, math supervisors, and…

This article describes experiences from a professional development project designed to prepare in-service eighth-grade mathematics teachers to develop, explore, and advance technological pedagogical content knowledge (TPCK) in the teaching and learning of Algebra I. This article describes the process of the participating teachers' mathematical…

At the heart of the recent focus on mathematics has been an increased emphasis on developing students' "number sense." Ironically, although growing as a force in the education literature, number sense has not been clearly defined for teachers. Teachers need specific support in understanding how to develop number sense in students, to…

In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a…

In this paper, we examine the potential use of animation for supporting the teaching of some of the mathematical concepts that underlie computer games development activities, such as vector and matrix algebra. An experiment was conducted with a group of UK undergraduate computing students to compare the perceived usefulness of animated and static…

Presents a model for teaching reasoning using technology in the contexts of algebra and geometry. (MKR)

This article describes a hands-on activity in which students convert decimals to fractions, measure, make scale drawings, use algebraic formulas, and use visualization to construct a three-dimensional physical model of a portion of the seafloor. In the process, they learn about sound, speed, and ocean exploration. (Contains 2 figures.)

Presents two methods for solving equations. An asymmetric approach works backward from a number by reversing operations performed on a variable. A symmetric approach views the equation as a scale and performs inverse operations on both sides of the balance to solve for the variable. (MDH)